It is as if Theresa May actively aspired to the kind of immortality conferrable by the popular internet meme: “You had one job.” Unless, given that he is said to run everything, credit should go to the cabinet secretary, Sir Jeremy Heywood. To avoid unfairness, maybe they should share with David Cameron the glory of having surpassed every variation on You Had One Jobsince its original manifestation in the 2001 film, Ocean’s Eleven, when a robber curses his dedicated alarm-disablers (a key alarm not having been disabled): “You tossers! You had one job to do!”

Either way, following the resignation, in July, of Dame Elizabeth Butler-Sloss from her chairmanship of the inquiry into historical child sex abuse, on account of awkward establishment connections, it was paramount for this team to pick a replacement whose loyalties were not in doubt.

Once the recommended legal/experience-related/great and good/worthy qualifications for running an inquiry into child abuse had been satisfied, May had to ensure that no further conflict of interest was perceptible to a public alert for institutional cover-ups, and to victims whose abuse was both inflicted and denied by the powerful. You might think that, above all, May had one job: to make sure the next chairman was not friends with Leon Brittan.

Lord Brittan being, of course, the former home secretary who was in office in the 80s when, it is widely believed, too little was done by the establishment to investigate child sexual abuse. A dossier compiled by the late MP Geoffrey Dickens, and passed by him to Lord Brittan, subsequently went missing, or was destroyed, a discovery that led to the announcement of this inquiry. Lord Brittan could well be prominent among its witnesses.

After lengthy consideration, May announced Butler-Sloss’s replacement. The new chairman would be Fiona Woolf. Woolf is the current lord mayor of London, a corporate lawyer, an eminent person, but not one with any recorded interest in child welfare or family law. She is also, we now learn, a friend of Lord Brittan. May had one job. What, unless it was a familiar pair of establishment hands, did she even see in Fiona Woolf, champion of City interests, that came close, even if she were guaranteed Brittan-free, to being a plausible qualification? Maybe, given this level of random, we should count ourselves lucky she didn’t pick Mary Berry or Guido Fawkes. Unless something has been lost in Home Office collaboration, the tone of an explanatory letter from Woolf could have been calculated to assure victims of abuse, who she sometime styles “the victim community”, that their interests remain as irrelevant to the powerful in 2014 as they were neglected by them in the past.

Avoiding the word friend, Woolf freely admits to “contacts” with the Brittans, of little consequence, she feels, since she has only had them three times to “a dinner party at my residence”, and twice been entertained “at their residence”, not forgetting some coffees with Mrs B – easily arranged since the families have lived in the same street since 2004.

Clearly, generations of lobbyists and PRs have got it quite wrong about the power of hospitality subtly to hobble and compromise; the 2010 Bribery Act went completely over the top.

Nor does it signify that the Brittans have also appeared times without number at the official boards, receptions, luncheons, banquets and dinners that give life meaning to the alderman community, even if, as Woolf insists, they do not make her part of the establishment, or even lodge in her mind. Her recollection of her most recent Brittan contact – in April 2013 – was promptly contradicted by a photograph taken at an awards ceremony six months later.

In defence of Woolf, the Times warns her critics not to underestimate the difficulty of finding an uncompromisable lawyer equipped to conduct a complex inquiry on this scale. That’s like an Italian expecting Silvio Berlusconi to ask for a birth certificate. And perhaps, for all the layperson knows, there really are no suitable candidates who do not live in the same street as the Brittans and have dinner with them once a year. Maybe, in the legal world, not being on dinner party terms with the Brittans is tantamount to gross negligence, to the point of disqualifying the non-compliant from the highest office.

But even so, given the complexity and scale of this inquiry, and bearing in mind the difficulty of tracking down the pure but also mentally alert, should we be troubled about its chairman’s fading powers of recall?

Did she, one member of the committee asked, send the Brittans a Christmas card. Again, Woolf was tested. “My Christmas card list last year had about 3,000 people on it. To be honest I don’t know whether they were on it or not.”

If, as her new colleagues must hope, this fomo (fear of missing out)-inducing behaviour lies within the normal range, there must be a case for the evolutionary psychologist Robin Dunbar reconsidering his famous number, currently fixed at 150 people, as the maximum with whom it’s possible to have meaningful relationships.

And should she ever feel the need to sacrifice her new position for the sake of public trust, Woolf must have a brilliant future as a one-woman dating agency.

As it is, she seems determined to stay, regardless of the victims’ wishes, and as much as this is catastrophic for the inquiry, whose panel should surely consider a boycott, you can almost appreciate her position. Are her own social connections so much more embarrassing than those exposed at the Leveson inquiry between senior politicians and Murdoch staffers who were entertaining, as well as the prime minister’s circle, senior policemen who would soon be investigating allegations of press hacking and corruption?

Cameron survived even though, unlike the circumspect Woolf, he had been happy to advertise Andy Coulson as his “friend”, ditto his partner in multiple texts and country suppers, Rebekah Brooks.

And it is not, Woolf might reasonably object, as if Leveson consigned such recklessness in office to the past, along with the web of old friends that Blair promoted. Honours are still sold, friends ennobled, MPs entertained by tobacco companies; the Department of Culture, Media and Sport recently saw nothing untoward in the BBC’s director general’s wife working for the company it paid to headhunt the next BBC chairman; for its part, the BBC was happy for its senior political interviewer, Andrew Marr, to launch his novel at a Downing Street party hosted by the prime minister.

The advance of networking as an ostensibly respectable activity is perfectly timed to absolve eminent practitioners of this continued carelessness about appearances, and to pass off their more dubious connections, forged, say, in the excitement of a Freud/Murdoch party, or at an auction of internships, as an estimable commitment to social plurality.

They are merely applying networking’s central precept: “It’s not what you know, but who you know.” Even, it turns out, in the field of child protection.